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Hamilton-Jacobi equations on networks
Yves Achdou, Fabio Camilli, Alessandra Cutri, Nicoletta Tchou
To cite this version:
Yves Achdou, Fabio Camilli, Alessandra Cutri, Nicoletta Tchou. Hamilton-Jacobi equations on net- works. IFAC Proceedings Volumes, Elsevier, 2011, 44 (1), pp.2577-2582. �10.3182/20110828-6IT- 1002.01084�. �hal-00503910v4�
Hamilton-Jacobi equations on networks
Yves Achdou ∗, Fabio Camilli †, Alessandra Cutr`ı‡, Nicoletta Tchou§ January 20, 2011
Abstract
We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. Under suitable assump- tions, we prove that the value function is continuous. We define a notion of constrained viscosity solution of Hamilton-Jacobi equations on the network and we study related com- parison principles. Under suitable assumptions, we prove in particular that the value function is the unique constrained viscosity solution of the Hamilton-Jacobi equation on the network.
Keywords Optimal control, graphs, networks, Hamilton-Jacobi equations, viscosity so- lutions
AMS 34H05, 49J15
1 Introduction
A network (or a graph) is a set of items, referred to as vertices or nodes, with connections between them referred to as edges. The main tools for the study of networks come from combinatorics and graph theory. But in the recent years there is an increasing interest in the investigation of dynamical systems and differential equation on networks, in particular in connection with problem of data transmission and traffic management (see for example Garavello-Piccoli [10], Engel et al [5]). In this perspective, the study of control problems on networks has interesting applications in various fields. Note that partial differential operators on ramified spaces have also been investigated, see e.g. [16], [15].
A typical optimal control problem is theminimum time problem, which consists of finding the shortest path between an initial position and a given target set. If the running cost is a fixed constant for each edge and the dynamics can go from one vertex to an adjacent one at each time step, the corresponding discrete-state discrete-time control problem can be studied via graph theory and matrix analysis. If instead the cost changes in a continuous way along the edges and the dynamics is continuous in time, the minimum time problem can be seen as a continuous-state continuous-time control problem where the admissible trajectories of the system are constrained to remain on the network. While control problems with state constrained in closures of open sets are well studied ([18, 19], [3], [11]) there is to our knowledge much fewer literature on problems in closed sets with empty interior. The results of Frankowska and Plaskacz [9, 8] do apply to some closed sets with empty interior, but not to networks with crosspoints (except in very particular cases).
∗Universit´e Paris 7, UFR Math´ematiques, 175 rue du Chevaleret, 75013 Paris, France, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France. achdou@math.jussieu.fr
†Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universit`a Roma ”La Sapienza” Via Antonio Scarpa 16, Italy, I-00161 Roma, camilli@dmmm.uniroma1.it
‡Dipartimento di Matematica, Universit`a ’Tor Vergata’ di Roma, 00133 Roma, Italy, cutri@mat.uniroma2.it
§IRMAR, Universit´e de Rennes 1, Rennes, France, nicoletta.tchou@univ-rennes1.fr
The aim of this paper is therefore to study optimal control problems whose dynamics is constrained to a network and the related Hamilton-Jacobi-Bellman equation. Note that other types of optimal control problems could be considered as well, leading to other boundary conditions at the endpoints of the network. In most of the paper, we will consider for simplicity the toy model given by a star-shaped network, i.e. straight edges intersecting at the origin. This simple model problem already contains many of the difficulties that we have to face in more general situations. Moreover we will sometimes assume that the running cost is independent of the control.
Since the dynamics is constrained to the network, the velocities tangent to the network vary from one edge to another, hence the set of the admissible controls depends on the state of the system. If the set of admissible controls varies in a continuous way, the cor- responding control problem can be studied via standard viscosity solution techniques (see Koike[12]). But for a network, the set of admissible controls drastically changes from a point in the interior of an edge, where only one direction is admissible (with possibly positive and negative velocities), to a vertex where the admissible directions are given by all the edges connected to it. Therefore, even if the data of the problem are regular, the correspond- ing Hamiltonian when restricted to the network has a discontinuous structure. Problem with discontinuous Hamiltonians have been recently studied by various authors (Tourin[23], Soravia[20], Deckelnick-Elliott[4], Bressan-Hong[2]), but the approaches and the results con- sidered in these papers do not seem to be applicable because of the particular structure of the considered domain. Finally, we very recently became aware of the thesis of Schieborn [17] devoted to the eikonal equation on networks, with an approach different from the one presented below.
Assuming that the set of the admissible control laws - i.e. the control laws for which the corresponding trajectory remains on the graph - is not empty, the control problem is well posed and the corresponding value function satisfies a dynamic programming principle. We introduce a first set of assumptions which guarantees that the value function is continuous on the network (with respect to the intrinsic geodetic distance).
The next step is to introduce a definition of weak solution which ensures the uniqueness of the continuous solution via a comparison theorem. While in the interior of an edge we can test the equation with a smooth test function, the main difficulties arise at the vertices where the network does not have a regular differential structure. At a vertex, we consider a concept of derivative similar to that of Dini’s derivative, see for example[1], hence regular test functions are the ones which admit derivatives in the directions of the edges entering in the node. We give a definition of viscosity solution on the network using the previous class of test functions. It is worthwhile to observe that this definition reduces to the classical one of viscosity solution if the graph is composed of two parallel segments entering in a node, see [1].
With this definition, the intrinsic geodetic distance, fixed one argument, is a regular function w.r.t. the other argument and it can be used in the comparison theorem as a penalty term in the classical doubling argument of viscosity solution theory.
We conclude observing that this paper is a first attempt to study Hamilton-Jacobi- Bellman equations and viscosity solutions on a network. Several points remain open such as more general control problems, problem with boundary conditions, stochastic control prob- lems.
The paper is organized as follows: the control problem and the basic assumptions are set in Section 2. In Section 3, we define useful notions and prove preliminary results, before proposing a definition of viscosity solutions of an Hamilton-Jacobi equation on the network in Section 4; then, we prove that this notion is equivalent to the classical one if the network is made of two parallel segments sharing one endpoint. We also prove that the value function of the control problem is a viscosity solution. Comparison principles are studied in Section 5.
Finally, in Section 6 we study a case when the value function may be discontinuous and we propose a notion of discontinuous viscosity solution.
2 Setting of the problem and basic assumptions
We consider a planar network with a finite number of edges and vertices. A network inR2 is a pair (V,E) where
i) V is a finite subset ofR2whose elements are said vertices
ii) E is a finite set of regular arcs ofR2, said edges, whose extrema are elements ofV. We say that two vertices are adjacent if they are connected by an edge. We say that a vertex belongs to ∂V (resp.,int(V)) if there is only one (resp., more than one) edge connected to it. We assume that the edges cross each other transversally. We denote by G the union of all the edges inE and all the vertices inV. We denote by G the setG\∂V.
Except when explicitly mentioned, we focus for simplicity on the model case of a star-shaped network withN straight edges, N >1, i.e.
G={O} ∪
N
[
j=1
Jj⊂R2, O= (0,0), Jj = (0,1)ej, (2.1) where (ej)j=1,...,N is a set of unit vectors in R2 s.t. ej 6=ek ifj 6=k. Note thatej =−ek
is possible. Then, ∂V = {ej, j = 1, . . . , N} and int(V) = {O}. We will use the notation
∂G ≡∂V. Except in§4.2, we assume that there is at least a pair (j, k),j6=ks.t. ej is not aligned withek.
The general case will be dealt with in a forthcoming paper, where we will also consider structures made of several manifolds of different dimensions crossing each other transversally.
Hereafter, the notationR+ stands for the interval [0,+∞).
For anyx∈ G, we denote byTx(G)⊂R2 the set of the tangent directions to the network, i.e.
p∈Tx(G) ⇐⇒ ∃T >0 andξ∈ C1([0, T];R2) s.t. ξ(t)∈ G,∀t≤T, ξ(0) =xand ˙ξ(0) =p.
(2.2) It is easy to prove thatp∈Tx(G) if and only if there exist sequences (tn)n∈N, tn >0 and (xn)n∈N,xn∈ G, such thattn →0+ and (xn−x)/tn→p.
We now introduce the optimal control problem onG. We start by making some assumptions on the structure of the problem.
Call B the closed unit ball of R2 centered at O. Take for A a compact set of R2 and a continuous functionf :B×A→R2 such that
|f(x, a)−f(y, a)| ≤L|x−y|, ∀x, y∈B, a∈A. (2.3) The assumption (2.3) implies that there existsM >0 such that
|f(x, a)| ≤M, ∀x∈B, a∈A. (2.4)
Additional assumptions will be made below. Forx∈ G, we consider the dynamical system y(t;˙ x, α) =f(y(t;x, α), α(t)), t >0,
y(0) =x. (2.5)
Remark 2.1. We have chosen to parametrize the dynamics by a functionf defined onB×A, i.e. on a much larger set than G ×A. We could also have defined f on G ×A only. This would have been equivalent since by Whitney extension theorem one can extend any Lipschitz function defined onGto a Lipschitz function defined onB. In fact, all the assumptions made below onf involvef|G×A only. Yet, it seemed to us that definingf onB×Aled to simpler notations.
Denoting by A the class of the control laws, i.e. the set of measurable functions from [0,+∞) toA, we introduce the subsetAx⊂ Aof the admissible control laws, i.e. the control laws for which the dynamics (2.5) is constrained on the networkG:
Ax={α∈ A: y(t;x, α)∈ G, ∀t >0}. (2.6)
Assumption 2.1.
Ax is not empty for any x∈ G. (2.7)
We will always considerα∈ Ax in (2.5).
We also define forx∈ G,
Ax={a∈A s.t. ∃θ >0 :y(t;x, a)∈ G,∀t,0< t < θ}. (2.8) From the continuity off, we see that for all a∈Ax,f(x, a)∈Tx(G).
Assumption 2.2. We assume that there exist non empty subsets Aj of A, j = 1, . . . , N, such that
Ax=Aj, ifx∈Jj, j= 1, . . . , N, (2.9) AO=∪Nj=1{a∈Aj:f(O, a)∈R+ej}, (2.10) Aej ={a∈Aj : f(ej, a)·ej ≤0} 6=∅, and inf
a∈Aejf(ej, a)·ej<0, j= 1, . . . , N.(2.11) and such thatAj=Ak ifej =−ek.
Remark 2.2. Assumption (2.9) says that the set of constant controls for which the trajecto- ries leavingx∈Jj stay inG for a positive time is nonempty and does not depend onx∈Jj. Assumption (2.10) characterizes the set of constant controls for which the trajectories leav- ingO stay inG for a positive time: a further assumption will be needed to state AO is not empty. The assumption in (2.11) at the vertices in∂V tells us that there exist controls which make the trajectory enter G; this assumption is classical in the context of state constrained problems.
Assumption 2.3. For allj∈1, . . . , N and allx∈Jj\{O}, there existsτ >0 such that for allα∈ Ax,α(t)∈Aj for almost allt∈[0, τ].
Assumption 2.3 says that for small durations, an admissible control law atxcannot take values outsideAx (except maybe on a negligeable set of times).
Assumption 2.4. We assume that there exist constants ζj >0 andζ
j >0,j = 1, . . . , N, s.t.
co(f(O, Aj)) = [−ζ
j, ζj]ej, ∀j= 1, . . . , N, (2.12) whereco(F)stands for the closed convex hull of F.
Remark 2.3. We will see that Assumption 2.4 implies the continuity of the value function, for which weaker assumptions can be made, see Remark 2.6.
Remark 2.4. Assumption 2.4 implies controllability near O.
Remark 2.5. Note that ifej =−ek then, from (2.12) and the continuity off,ζj =ζ
k and ζk=ζj.
Example 2.1. Take forA the unit ball of R2 and f(x, a) = g(x)a where g : B → R is a Lipschitz continuous positive function: we can see that all the assumptions above are satisfied.
In particular, let us show that Assumption 2.3 holds in the present case: take x∈ G\{O}, for example x ∈ J1 and α ∈ Ax. With M as in (2.4), take τx =|x|/(2M). It is easy to see that y(t;x, α)∈J1 fort∈[0, τx]. This implies that Rt
0e1∧f(y(s;x, α), α(s))ds= 0 for t ∈ [0, τx], and therefore e1∧f(y(t;x, α), α(t)) = g(y(t;x, α))e1∧α(t) = 0 for almost all t∈[0, τx]. Therefore, sinceg is positive,α(t)∈A1=A∩Re1=Axfor almost allt∈[0, τx].
Example 2.2. Take N unit vectors (ej)j=1,...,N, with ej = (cosθj,sinθj), θj ∈ [0,2π).
Chooseζj,ζ
j 2N positive numbers such thatζj =ζ
k andζk =ζ
j ifej=−ek. Take forAthe unit ball ofR2; letζ:R→R+ be a2π-periodic and continuous function such thatζ(θj) =ζj and ζ(−θj) =ζj, j = 1, . . . , N; Choose f(x, a) =g(x)ζ(θ)a where a=|a|(cosθ,sinθ) and g:B →R is a Lipschitz continuous positive function. We can see that all the assumptions above are satisfied.
Example 2.3. Choose N unit vectors (ej)j=1,...,N and 2N positive numbers ζj, ζ
j as in Example 2.2. TakeA=∪Nj=1Kej,K={−1,1}. Choose
f(x, a) =g(x)
N
X
j=1
−ζ
j1a=−ej +ζj1a=ej
ej
where g : B → R is a Lipschitz continuous positive function. We can see that all the assumptions above are satisfied.
Example 2.4. As a particular case of Example 2.3, one may take the cross shaped network G = {O} ∪S4
j=1Jj, J1 = (0,1)e1, J2 = −(0,1)e1, J3 = (0,1)e2, J4 = −(0,1)e2, e1 and e2 being two orthogonal unit vectors. One may chooseA =Ke1∪Ke2, K = {−1,1} and f(x, a) =g(x)a whereg:B→Ris a Lipschitz continuous positive function.
Finally, we consider a continuous functions`:B×A→Rsuch that
|`(x, a)| ≤M, ∀x∈B, a∈A, (2.13)
|`(x, a)−`(y, a)| ≤L|x−y|, ∀x, y∈B, a∈A. (2.14) Forλ >0, we consider the cost functional
J(x, α) = Z ∞
0
`(y(t;x, α), α(t))e−λtdt. (2.15) The value function of the constrained control problem on the network is
v(x) = inf
α∈Ax
J(x, α), x∈ G. (2.16)
Assumption 2.1 and the assumptions on`are enough for the dynamic programming principle:
v(x) = inf
α∈Ax
Z t 0
`(y(s;x, α), α(s))e−λsds+e−λtv(y(t;x, α))
. (2.17)
The proof is standard along the arguments in Propositions III.2.5 or IV.5.5 in [1].
Proposition 2.1. Under the assumptions above, the value function is continuous onG.
Proof. For the continuity atx∈∂V, we refer to [1], proof of Theorem 5.2, page 274. We are going to study the continuity of the value function atx∈ G.
Consider now x ∈ G. We want to prove that lim supz→xv(z) ≤ v(x). The inequality lim infz→xv(z)≥v(x) is obtained in a similar way.
For anyε >0, there exists a controlα∈ Ax such thatJ(x,α)¯ < v(x) +ε.
The following observation will be useful: from the controllability assumption (2.12) and from (2.3), there exist a positive numberr0and a constantCsuch that for allz1, z2∈BO(r0)∩ G, there existsαz1,z2∈ Az1 andτz1,z2 ≤C|z1−z2|withy(τz1,z2;z1, αz1,z2) =z2.
We distinguish two cases: a)x∈BO(r0/2); b)x /∈BO(r0/2).
a) Ifx∈BO(r0/2), then ifz∈BO(r0), we construct ˜α∈ Az as follows:
˜
α(t) = αz,x(t) ift < τz,x,
˜
α(t) = α(t−τz,x) ift > τz,x.
Sinceτz,x≤C|z−x|, it is easy to prove thatv(z)≤J(z,α)˜ ≤v(x) +ε+CM|x−z|. Sending εto 0, we obtain that lim supz→xv(z)≤v(x) forx∈BO(r0/2).
b) Ifx /∈ BO(r0/2), we can assume that x∈J1. Then we can choose z close enough to x such thatz belongs toJ1.
Therefore, the controlαis also admissible for zat least for a finite duration, (the first time T wheny(t;x, α) ory(t;z, α) hitsO ore1, if it exists). For brevity, we will only discuss the
case when y(T;x, α) = O or y(T;z, α) =O, if T exists. The other cases y(T;x, α) =e1 or y(T;z, α) = e1 can be dealt with in a similar way by using the controllability assumption (2.11), see [1] for example.
We define
Tε=−1
λlog(ελ/(2M)), Cε=L Z Tε
0
e(L−λ)tdt, ρε=r0e−LTε/4, (2.18) and we can always assume that|x−z| ≤ρε.
b1) If T > Tε or T does not exist, then both y(t;z, α) and y(t;x, α) remain in J1∪ {e1} fort < Tε. For any ˜α∈ Az s.t. ˜α(t) =α(t) for t < Tε, we have that|J(z,α)˜ −J(x,α)| ≤˜ Cε|x−z|+ε, whereCεis defined in (2.18). Thusv(z)≤J(z,α)˜ ≤v(x) +Cε|x−z|+ 3ε.
b2) Ify(T;x, α) =O, then we construct the control ˜α∈ Azas follows
˜
α(t) = α(t) ift < T,
˜
α(t) = αy(T;z,α),O(t−T) ifT < t < T +τy(T;z,α),O,
˜
α(t) = α(t−τy(T;z,α),O) ift > T +τy(T;z,α),O.
Note that this is possible since|x−z| ≤ρε which implies|y(T;z, α)| ≤eLTε|x−z| ≤r0/4.
Here again, we get that
v(z)≤J(z,α)˜ ≤v(x) + ˜Cε|x−z|+ε, for another constant ˜Cε.
b3) Ify(T;z, α) =O, then we construct the control ˜α∈ Az as follows
˜
α(t) = α(t) ift < T,
˜
α(t) = αO,y(T;x,α)(t−T) ifT < t < T +τO,y(T;x,α),
˜
α(t) = α(t−τO,y(T;x,α)) ift > T +τO,y(T;x,α).
Note that this is possible since|x−z| ≤ρεwhich implies|y(T;x, α)| ≤eLTε|x−z| ≤r0/4.
Here again, we get that
v(z)≤J(z,α)˜ ≤v(x) + ˜Cε|x−z|+ε.
u t
Remark 2.6. It can be shown that Proposition 2.1 holds if for some indicesj,ζj =ζj = 0.
We now give an example in which the value function is discontinuous: let (e1, e2) be an orthonormal basis ofR2,G= (0,1)e1∪ {O} ∪(0,1)e2,A={0, e1, e2},f(x, a) =a(1−2|x|).
Take`(x, a) = 1 ifx2= 0 and`(x, a) = 1− |x|ifx1= 0. Assumption 2.4 is not satisfied. It is easy to compute the value functionvat x= (x1, x2): we have
v(x1,0) = 1
λ, 0< x1≤1, v(0, x2) = 1
2λ+1−2x2
4 + 2λ, 0≤x2< 1 2, v(0, x2) = 1−x2
λ , 1
2 ≤x2≤1.
(2.19)
The value function is discontinuous atO.
3 Preliminary notions for weak solutions
Hereafter, we make all the assumptions stated in§2, except in§4.2 and§6. All the theorems below will be stated without repeating the assumptions.
3.1 Test functions
We introduce the class of the admissible test functions for the differential equation on the network
Definition 3.1. We say that a function ϕ: G →R is an admissible test function and we writeϕ∈ R(G)if
• ϕis continuous in G andC1 inG \ {O}
• for anyj,j = 1, . . . , N,ϕ|J
j ∈ C1(Jj).
Therefore, for any ζ ∈R2 such that there exists a continuous function z: [0,1]→ G and a sequence (tn)n∈N,0< tn ≤1 with tn→0 and
n→∞lim z(tn)
tn =ζ, the limitlimn→∞ϕ(z(tn))−ϕ(O)t
n exists and does not depend onz and(tn)n∈N. We define Dϕ(O, ζ) = lim
n→∞
ϕ(z(tn))−ϕ(O) tn
. (3.1)
If x∈ G\{O} andζ∈Tx(G), we agree to write Dϕ(x, ζ) =Dϕ(x)·ζ.
Property 3.1. Let us observe thatDϕ(O, ρζ) =ρDϕ(O, ζ)for anyρ >0. Indeed, denoting byτn=tn/ρ,limn→∞z(tn)/τn=ρζ. Hence,
ρDϕ(O, ζ) = lim
n→∞
ϕ(z(tn))−ϕ(O)
τn =Dϕ(O, ρζ).
As shown below, this property is not true ifρ <0.
Ifϕ∈ C1(R2), thenϕ|G∈ R(G) andDϕ(O, ζ) =Dϕ(O)·ζfor anyζ∈R+ej,j= 1, . . . , N. Ifej=−ek for somej6=k∈ {1, . . . , N},Dϕ(O, ej) =−Dϕ(O,−ej).
Ifϕis continuous and ϕ|G∩¯ Rej is C1 for j= 1, . . . , N, thenϕ∈ R(G) but the converse may not be true if two edges are aligned: for example, ifej =−ek for somej 6=k∈ {1, . . . , N}, the functionx7→b|x|belongs toR(G) andDϕ(O, ej) =Dϕ(O,−ej) =b.
Property 3.2. Ifϕ=g◦ψ withg∈ C1 andψ∈ R(G), thenϕ∈ R(G)and
Dϕ(O, ζ) =g0(ψ(O))Dψ(O, ζ). (3.2)
3.2 A set of relaxed vector fields
Let us use the notation
mO= min
a∈∪1≤k≤NAk
`(O, a). (3.3)
We will sometimes make a further assumption:
Assumption 3.1. The function `:G ×A→Rsatisfies: for all j= 1, . . . , N, (0, mO)∈co (f(O, a), `(O, a)) :a∈Aj
. (3.4)
Note that from Assumption 2.4, for all j= 1, . . . , N, 0∈co (f(O, a) :a∈Aj .
Example 3.1. From Assumption 2.4, Assumption 3.1 is always satisfied if `(O, a) does no depend ona.
Example 3.2. In the examples 2.1- 2.4, we can take`(x, a) =q(x)|a|ν+p(x), whereν≥0 andqand pare Lipschitz functions defined on G withq(O)≥0.
Definition 3.2. Forx∈ G, we introduce the sets
f˜(x) =
η∈Tx(G) : ∃(αn)n∈N, αn∈ Ax,
∃(tn)n∈N s.t.
tn →0+and
n→∞lim 1 tn
Z tn
0
f(y(t;x, αn), αn(t))dt=η
and
f`(x) =e
(η, µ)∈Tx(G)×R: ∃(αn)n∈N, αn∈ Ax,
∃(tn)n∈N s.t.
tn→0+,
n→∞lim 1 tn
Z tn
0
f(y(t;x, αn), αn(t))dt=η,
n→∞lim 1 tn
Z tn 0
`(y(t;x, αn), αn(t))dt=µ
.
Proposition 3.1. a) Under all the assumptions made in §2,
f`(x) =e FL(x) ≡co ((f(x, a), `(x, a)) :a∈Ax), ifx∈ G\{O}, (3.5) f`(O)e ⊃ FL(O) ≡ N∪
j=1
co (f(O, a), `(O, a)) :a∈Aj
∩(R+ej×R)
, (3.6) f`(ee j) = FL(ej) ≡co (f(ej, a), `(ej, a)) :a∈Aj
∩(R−ej×R). (3.7) b) Under all the assumptions made in§2 and Assumption 3.1,
1. For allζ∈f˜(O)∩R+ej, there existsξ∈Rsuch that(ζ, ξ)∈FL(O)∩(R+ej×R).
2. For allζ∈f˜(O),
min{µ: (ζ, µ)∈FL(O)}= minn
µ: (ζ, µ)∈f`(O)e o
. (3.8)
Proof. Take firstx∈ G\{O}.
We can assume that x ∈ J1. The inclusion FL(x) ⊂ f`(x) is obtained as follows: takee ζ=PJ
j=1µjf(x, aj),ξ=PJ
j=1µj`(x, aj) withaj∈AxandP
jµj = 1, 0≤µj. Fortnsmall enough, it is possible to construct a controlαn ∈ Axsuch thatαn(t) =aj for (P
k<jµk)tn<
t ≤ (P
k≤jµk)tn: we have t1
n
Rtn
0 f(y(t;x, αn), αn(t))dt = t1
n
Rtn
0 f(x, αn(t))dt +o(1) = P
jµjf(x, aj) +o(1), so
n→∞lim 1 tn
Z tn 0
f(y(t;x, αn), αn(t))dt=ζ.
Similarly,
n→∞lim 1 tn
Z tn 0
`(y(t;x, αn), αn(t))dt=ξ.
Finally, for (ζ, ξ)∈FL(x), we approximate (ζ, ξ) by (ζm, ξm)m∈N, where (ζm, ξm) is a convex combination of (f(x, a), `(x, a)),a∈Ax, and we conclude by a diagonal process.
For the opposite inclusion, sincex∈ G\{O}, we know from Assumption 2.3 that there exists τ >0, such that for allα∈ Ax,α(t)∈Axfor 0≤t < τ. Therefore,
1 s
Z s 0
f(x, α(t))dt,1 s
Z s 0
`(x, α(t))dt
∈FL(x)
forssmall enough. This and the Lipschitz continuity off and` w.r.t. their first argument imply thatf`(x)e ⊂FL(x). We have proved (3.5).
We now consider x = O. We first discuss the inclusion FL(O) ⊂ f`(O): we takee ζ = PJ
j=1µjf(O, aj), ξ=PJ
j=1µj`(O, aj) with aj ∈A1 and we assume that ζ ∈R+e1. Up to a permutation of the indices, it is possible to assume that there existsJ0, 1< J0 ≤J such
that f(O, aj)∈R+e1 forj ≤J0 and that f(O, aj) ∈R−e1 for j > J0. Then by a similar argument as above, (ζ, ξ)∈f`(O). By a diagonal process, this implies thate
co (f(O, a), `(O, a)) :a∈A1
∩(R+e1×R)⊂f`(O).e Similarly co (f(O, a), `(O, a)) :a∈Aj
∩(R+ej×R)⊂f`(O), so we have proved (3.6).e The proof of (3.7) is similar.
To prove points b 1) and b 2), we considerζ∈f˜(O) and make out two cases:
•ζ= 0: from Assumption 2.4, FL(O)∩({0} ×R)6=∅.
From Assumption 3.1, min{ξ: (0, ξ)∈FL(O)}=mO. On the other hand, for all sequencestn→0+ andαn ∈ AO,
lim inf
n→∞
1 tn
Z tn 0
`(y(t;O, αn), αn(t))dt≥mO. Therefore,
min{ξ: (0, ξ)∈FL(O)} ≤minn
ξ: (0, ξ)∈f`(O)e o , and this inequality is in fact an identity, because FL(O)⊂f`(O).e
• ζ 6= 0: we can suppose that 0 6=ζ ∈ R+e1. There exist sequences αn ∈ AO and tn >0 such thattn→0+, limn→∞t1
n
Rtn
0 f(y(t;O, αn), αn(t))dt=ζ. Up to an extraction, we may assume that limn→∞t1
n
Rtn
0 `(y(t;O, αn), αn(t))dt=µ.
Since 06=ζ∈R+e1, there existssn, 0≤sn< tnsuch thaty(sn;O, αn) =Oandy(t;O, αn)∈ J1 for all t, sn < t ≤ tn. From Assumption 2.3, this implies that αn(t) ∈ A1 for all t, sn< t < tn. Hence,
1 tn−sn
Z tn sn
f(O, αn(t))dt, 1 tn−sn
Z tn sn
`(O, αn(t))dt
∈co (f(O, a), `(O, a)) :a∈A1
∩(R+e1×R).
Therefore, since (0, mO)∈co (f(O, a), `(O, a)) :a∈A1
from Assumption 3.1, we get that 1
tn
Z tn sn
f(O, αn(t))dt, 1 tn
Z tn sn
`(O, αn(t))dt+sn
tn
mO
∈co (f(O, a), `(O, a)) :a∈A1
∩(R+e1×R).
Up to the extraction of a subsequence, we may say that t1
n
Rtn
sn `(O, αn(t))dt+stn
nmOconverges to a real numberξ. Moreover, from the Lipschitz continuity off,
ζ= lim
n→∞
1 tn
Z tn sn
f(y(t;O, αn), αn(t))dt= lim
n→∞
1 tn
Z tn sn
f(O, αn(t))dt, and we see that (ζ, ξ)∈FL(O)∩(R+e1×R), which proves point b 1).
We also see that
ξ≤lim 1 tn
Z tn 0
`(O, αn(t))dt=µ,
where the last identity comes from the Lipschitz continuity of`. We have proved point b 2), sinceξ≤µis true for allµsuch that (ζ, µ)∈f`(O).e ut
4 Viscosity solutions
Hereafter, unless explicitly mentioned, we make all the assumptions of§2.
